Perfect imperfection, imperfections of our understanding, or…

If pi, 3.14…, is the ratio reached when dividing the circumference of a circle by its diameter, how then is the result of this quotient an irrational number and not a quotient? The main quality of an irrational number I’m addressing here is that the left side of the decimal results in an infinite non-repeating sequence of numbers. Rational numbers, which all quotients of integers are, have either terminating or repeating decimals.

The problem is that if you take the circumference and straighten it out into a line, and then compare that length to the length of the diameter, the two lengths remain incommensurable. They can share no common measure. Meaning no matter how small you make a unit of measure, that measure will never produce a whole number value for both lines. This means one length must begin as an irrational number, which seems unnatural or inconsistent with how one initially thinks of “number” to be represented in the world.

This is not a problem unique to circles. Any geometric shape, either initially or when subdivided as a composition of other geometric shapes, will result in an incommensurable ratio between two or more sides. Although, pi does take things one step further, in that it is a transcendental number, but one thing at a time. What I’m currently contemplating, what does this suggest?